I'm starting with the caveat that I haven't put the thought into this (yet) that OP has, so my ideas are still developing and many might be better described as "hunches." In other words, I might be completely wrong. I probably am. It happens a lot.
In my gut, a d20 system (or a D100 system, even more so) signifies a "rules heavy" game. These are games that tend to want to offer lots of granular detail covering different choices that players might make in the game. There are lots of tables, for everything from weapons to spells to random encounters. AD&D even had a table for random prostitute characteristics (I am not making that up; I guess this was a big gameplay issue for Gygax and friends). In general, I associate d20 games with rules that want to offer more constraints on how the game is to be played.
D6 games tend to be more built around a specific design concept, and often are boutique games. I don't associate them with long campaigns. They are often very thematic, and can be extremely inventive with imagining new ways to generate role-play situations and player interactions. Some even challenge the basic construction of the GM/Player dynamic, such as the first edition of Fiasco (a game I still adore). Others are tailored to very specific settings and have some elements of a board game to them.
For Derek, I immediately think of my favourite (?) RPG, Dread (I know, I know, I bring it up all the time). It gets rid of dice altogether and instead uses a jenga tower. That totally changes the structure of roleplaying by making the slow build of narrative tension the centrepiece of the game. Ten Candles is a D6 game that similarly emphasizes dramatic structure over complicated rules.
So my initial, possibly stupid, thoughts are that complicated dice (d20 systems as defined by OP) tend to go with complicated games where rules are very important and the emphasis is on long campaigns, and simpler dice (d6 as defined by OP) tend to go with simpler games that emphasize one-off games. D20 games are about designer or DM control, D6 games about player freedom (and neither of those is necessarily better).
Edit: also, I tend to think of d20 games as all D&D, just with different skins on. By which I mean that the game play experience is always basically similar in terms of the relationship between the GM, players, and rules.